Lab IV · Crane hoist braking and trolley anti-sway

Objectives

As set by the Lab Works booklet:

1. Simulate the four-quadrant torque and power of a crane hoist raising and lowering a rated load.
2. Size a dynamic-braking resistor from the simulated peak regenerated power.
3. Model the suspended load as a simple pendulum and simulate its sway response to a trolley acceleration profile.
4. Design and simulate a shaped (anti-sway) acceleration profile and compare its residual sway with an unshaped (step) profile.

Background

Chapter 4 established that hoist torque follows $T_\text{hoist} = mgR / (i\,\eta^{\pm 1})$ (course eq. 4.1, with the motoring and generating exponent of Chapter 3), and that a load on a rope of length $L$ is a pendulum of natural period $T = 2\pi\sqrt{L/g}$ (course eq. 4.5), excited by the trolley acceleration. For small angles the sway angle $\theta$ under a trolley acceleration $a(t)$ obeys (eq. 4.1 of the Lab Works booklet):

$$\ddot\theta + \frac{g}{L}\,\theta = -\frac{a(t)}{L}$$

Anti-sway control shapes $a(t)$, for example with two acceleration pulses spaced by half the pendulum period ("input shaping"), so that the residual oscillation after the trolley reaches constant speed is cancelled.

Parameter sheet

Parameter	Symbol	Value
Rated hoist load	$m$	5000 kg
Hoisting speed	$v$	0.25 m/s
Drum radius	$R$	0.4 m
Mechanical efficiency	$\eta$	0.85
DC-link voltage	$U_\text{dc}$	620 V
Rope (sling) length to load	$L$	4 m
Trolley target speed	$v_\text{trolley}$	0.5 m/s
Trolley acceleration time	$t_a$	2 s

Procedure

The booklet's five steps, with the names the checker looks for shown in code font.

1. Set up the environment. Import NumPy and Matplotlib (already in the template).
2. Define parameters. Set m, v, R, eta, Udc, L, v_trolley and t_a.
3. Implement the hoist model. Compute P_load and P_motor for raising, the gear ratio i_gear, the motor torques T_raise and T_lower, the power P_gen returned when lowering the same load at the same speed, and the braking resistor R_brake.
4. Implement the pendulum model. Write simulate_sway(L, accel, t_end, dt=0.01) returning the arrays t, theta for an acceleration function accel(t). Integrate with RK4 or scipy.integrate.solve_ivp: plain explicit Euler slowly pumps energy into an undamped oscillator. Then define step_accel (unshaped), T_sway, and shaped_accel (your anti-sway profile), and compute residual_unshaped and residual_shaped: the largest $|\theta|$ once the trolley has stopped accelerating.
5. Plot and analyse. Plot the hoist torque and power for the raise and lower cycle, and $\theta(t)$ for both trolley profiles, then answer the questions under the workspace.

The checker calls simulate_sway with a random rope length and acceleration step on every run, so it must work for any sensible input.

Report

One individual report and your code, in this order: Objective, Model, Parameters, Results, Discussion, Conclusion, Code, marked out of 20 (model 6, results and plots 5, discussion 6, report quality 3). The workspace builds it from your work.